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<H3><A NAME="SECTION03235100000000000000"></A>
<A NAME="secGSEP"></A>
<BR>
Generalized Symmetric Definite Eigenproblems (GSEP)
</H3>

<P>
<A NAME="1695"></A><A NAME="1696"></A>
Drivers are provided to compute all the eigenvalues and
(optionally) the eigenvectors of the following types of problems:

<P>
<DL COMPACT>
<DT>1.
<DD>
<!-- MATH
 $A z = \lambda B z$
 -->
<IMG
 WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img39.gif"
 ALT="$A z = \lambda B z$">
<P>
<DT>2.
<DD>
<!-- MATH
 $A B z = \lambda z$
 -->
<IMG
 WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img40.gif"
 ALT="$A B z = \lambda z$">

<P>
<DT>3.
<DD>
<!-- MATH
 $B A z = \lambda z$
 -->
<IMG
 WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img41.gif"
 ALT="$B A z = \lambda z$">

<P>
</DL>

<P>
where <I>A</I> and <I>B</I> are symmetric or Hermitian and <I>B</I> is positive definite.
For all these problems the eigenvalues<A NAME="1699"></A> <IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img23.gif"
 ALT="$\lambda$">
are real. The matrices <I>Z</I>
of computed eigenvectors<A NAME="1700"></A> satisfy

<!-- MATH
 $Z^T A Z = \Lambda$
 -->
<IMG
 WIDTH="90" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img42.gif"
 ALT="$Z^T A Z = \Lambda$">
(problem types 1 and 3) or 
<!-- MATH
 $Z^{-1} A Z^{-T} = I$
 -->
<I>Z</I><SUP>-1</SUP> <I>A Z</I><SUP>-<I>T</I></SUP> = <I>I</I>
(problem type 2), where <IMG
 WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img28.gif"
 ALT="$\Lambda$">
is a diagonal matrix with the eigenvalues
on the diagonal. <I>Z</I> also satisfies
<I>Z</I><SUP><I>T</I></SUP> <I>B Z</I> = <I>I</I> (problem types 1 and 2) or 
<!-- MATH
 $Z^T B^{-1} Z = I$
 -->
<I>Z</I><SUP><I>T</I></SUP> <I>B</I><SUP>-1</SUP> <I>Z</I> = <I>I</I> (problem type 3).

<P>
There are three types of driver routines for generalized symmetric and
Hermitian eigenproblems.  Originally LAPACK had just the simple and expert
drivers described below, and the other one was added after an improved algorithm
was discovered.

<P>

<UL><LI>a <B>simple</B> driver (name ending -GV)<A NAME="1706"></A>
      computes all the eigenvalues and (optionally) eigenvectors.

<P>

<LI>an <B>expert</B> driver
      (name ending -GVX)<A NAME="1708"></A> computes
      all or a selected subset of the eigenvalues and (optionally) eigenvectors.
      If few enough eigenvalues or eigenvectors are desired, the expert driver
      is faster than the simple driver.

<P>

<LI>a <B>divide-and-conquer</B> driver
      (name ending -GVD)<A NAME="1710"></A> solves the
      same problem as the simple driver. It is much faster than the simple
      driver for large matrices, but uses more workspace. The name
      divide-and-conquer<A NAME="1711"></A> refers to the underlying
      algorithm (see sections <A HREF="node48.html#subseccompsep">2.4.4</A> and <A HREF="node70.html#subsecblockeig">3.4.3</A>).

<P>

</UL>

<P>
Different driver routines are provided to take advantage of special
structure or storage of the matrices <I>A</I> and <I>B</I>, as shown in
Table&nbsp;<A HREF="node36.html#tabdrivegeig">2.6</A>.

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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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